3.1080 \(\int \frac{x^2}{(-2 a-b x^2) (-a-b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}} \]

[Out]

ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2)) - ArcTanh[(Sqrt[b]*x)/(Sqrt
[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2))

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Rubi [A]  time = 0.0332707, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((-2*a - b*x^2)*(-a - b*x^2)^(3/4)),x]

[Out]

ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2)) - ArcTanh[(Sqrt[b]*x)/(Sqrt
[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2))

Rule 442

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> -Simp[(b*ArcTan[(Rt[-(b^2/a), 4]*
x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] + Simp[(b*ArcTanh[(Rt[-(b^2/a), 4]*x)/(Sq
rt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0
] && NegQ[b^2/a]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0589936, size = 70, normalized size = 0.71 \[ -\frac{x^3 \left (\frac{a+b x^2}{a}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{6 a \left (-a-b x^2\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/((-2*a - b*x^2)*(-a - b*x^2)^(3/4)),x]

[Out]

-(x^3*((a + b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -(b*x^2)/(2*a)])/(6*a*(-a - b*x^2)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-b{x}^{2}-2\,a} \left ( -b{x}^{2}-a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x)

[Out]

int(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="maxima")

[Out]

-integrate(x^2/((b*x^2 + 2*a)*(-b*x^2 - a)^(3/4)), x)

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Fricas [B]  time = 1.67294, size = 559, normalized size = 5.7 \begin{align*} 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{4 \,{\left (\sqrt{\frac{1}{2}} \left (\frac{1}{4}\right )^{\frac{3}{4}} a b^{4} x \sqrt{\frac{b^{4} x^{2} \sqrt{\frac{1}{a b^{6}}} + 2 \, \sqrt{-b x^{2} - a}}{x^{2}}} \left (\frac{1}{a b^{6}}\right )^{\frac{3}{4}} - \left (\frac{1}{4}\right )^{\frac{3}{4}}{\left (-b x^{2} - a\right )}^{\frac{1}{4}} a b^{4} \left (\frac{1}{a b^{6}}\right )^{\frac{3}{4}}\right )}}{x}\right ) - \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} +{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} -{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="fricas")

[Out]

2*(1/4)^(1/4)*(1/(a*b^6))^(1/4)*arctan(4*(sqrt(1/2)*(1/4)^(3/4)*a*b^4*x*sqrt((b^4*x^2*sqrt(1/(a*b^6)) + 2*sqrt
(-b*x^2 - a))/x^2)*(1/(a*b^6))^(3/4) - (1/4)^(3/4)*(-b*x^2 - a)^(1/4)*a*b^4*(1/(a*b^6))^(3/4))/x) - 1/2*(1/4)^
(1/4)*(1/(a*b^6))^(1/4)*log(((1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) + (-b*x^2 - a)^(1/4))/x) + 1/2*(1/4)^(1/4)*(1
/(a*b^6))^(1/4)*log(-((1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) - (-b*x^2 - a)^(1/4))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{2 a \left (- a - b x^{2}\right )^{\frac{3}{4}} + b x^{2} \left (- a - b x^{2}\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-b*x**2-2*a)/(-b*x**2-a)**(3/4),x)

[Out]

-Integral(x**2/(2*a*(-a - b*x**2)**(3/4) + b*x**2*(-a - b*x**2)**(3/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="giac")

[Out]

integrate(-x^2/((b*x^2 + 2*a)*(-b*x^2 - a)^(3/4)), x)